If you want to divide the pre-history of humanity into chunks, one of the common ways to do it is to break it up into (1) the period before modern humans evolved and after our ancestors split off from chimpanzees and bonobos (the pre-human era), (2) the period from the appearance of Neanderthals until the appearance of modern humans (the Neaderthal era), (3) the period from the appearance of modern humans until the "Neolithic Revolution," (modern human pre-history) and (4) the period after the Neolithic Revolution (human history a.k.a. the Holocene period).
The Neolithic Revolution, approximately 8,000-12,000 years ago represents the time period in which humans transitioned from being hunter-gatherers to farmers, writing was developed, and cities first arose. The oldest linguistic evidence about our roots from language family resemblances also goes back only about this far (the vast linguistic diversity of pre-Columbian America also emerged over a time period only a few thousand years older, at most).
Modern humans, the subspecies distinct from Neanderthals to which every human alive today belongs, arose sometime roughly 160,000 to 100,000 years ago, and spread across the globe over tens of thousands of years, with the Americas constituting the areas last settled by humans, other than a few islands in Oceana. Humans had dispersed to every continent but Antarctica by the time of the Neolithic Revolution, which occurred everywhere on Earth at roughly the same time (give or take a few thousand years), with different plants and animals being domesticated, and different architectural styles being used in different places.
Our understanding of modern human pre-history is pretty limited. Illiterate hunter-gatherers don't leave a lot of traces behind for archeologists to examine. We know that Neanderthals went extinct, or nearly extinct, sometime fairly late in modern human pre-history. We can use genetic evidence to trace some very broad patterns of migration of humans that probably took place in this time period.
Natural Climate Change and Human Pre-History
The Neolithic Revolution coincides with the end of the most recent Ice Age.
The latest plausible evidence of Neanderthals dates to about 24,000 years ago, around the same time as the coldest point of the most recent ice age, and also the coldest time period since modern humans emerged from Africa.
The emergence of modern humans from Africa roughly coincides with the interglacial Late Pleistocene Eemian Stage, 131–114 thousand years ago, whose end marked the beginning of the Ice Age that ended shortly before the Neolithic Revolution.
And, there are two interglacial periods that could coincide neatly with the emergence of Neaderthals from Africa sometime 350,000-600,000 ago.
It is plausible to hypothesize that the most recent ice age tipped the balance that caused Neaderthals to go exinct and be replaced entirely by modern humans, and that the setbacks suffered by Neanderthals in the previous ice age is what created the ecological space into which modern humans emerged from Africa as that ice age ended.
Domesticated Dogs, Domesticated Cats and the Neolithic Revolution
There is a near consensus based upon genetic evidence that dogs were domesticated in modern human pre-history, with some genetic evidence pointing to 100,000 years ago, around the time that modern humans left Africa, and other genetic evidence pointing to a much later date, around 15,000 years ago, a few thousand years before the Neolithic Revolution. Either way, dogs would be the first domesticated animals.
The notion of dogs as the first domesticated animals is consistent with the fact that pre-Columbian Native Americans had domesticated dogs, but do not appear to have brought other domesticated farm animals with them to the New World from Asia (in contrast, modern human settlers of the last uninhabited islands in the Pacific did bring domesticated animals with them). The animals domesticated in the New World during the Neolithic Revolution were indigeneous to the Americas. This is also consistent with the fact that many Native American lived on a hunter-gatherer basis in pre-Columbian times, even though there were many notable farming based communities in the Americas.
Indeed, the coincidence in time between the later dates for the domestication of dogs, a theory that points to an origin for all domesticated dogs somewhere in China, is suggestive of the idea that the domestication of dogs may have been a pivotal cause of the entire Neolithic Revolution.
Dogs may have been important both in suggesting to pre-modern humans the benefits that could come from domesticating animals, and also in the actual proces of farming by protecting crops and domesticated animals and the human settlements themselves from invaders. The domestication of dogs, along with an end to competition between modern humans and Neanderthals after Neaderthals were wiped out during the ice age, and the improved climate for growing plants that arrived when the ice age ended, may have been a critical factor that made agriculture a sustainable proposition.
Cats, in contrast, appear to have been domesticated several thousand years later than dogs, probably co-existing with human communities only after the food surpluses created by human adoption of agriculture started to attract the vermin that cats prey upon. Genetic evidence suggests that all domesticated cats trace their roots to Egypt. Domesticated cats are not native to the Americas (or to either New Zealand or Australia where their introduction caused the extinction or near extinction of many native marsupial species). According to Wikipedia:
In 2004, a grave was excavated in Cyprus that contained the skeletons, laid close to one another, of both a human and a cat. The grave is estimated to be 9,500 years old, pushing back the earliest known feline-human association significantly.
Like some other domesticated animals, cats live in a mutualistic arrangement with humans. It is believed that the benefit of removing rats and mice from humans' food stores outweighed the trouble of extending the protection of a human settlement to a formerly wild animal, almost certainly for humans who had adopted a farming economy. Unlike the dog, which also hunts and kills rodents, the cat does not eat grains, fruits, or vegetables.
In modern rural areas, farms often have dozens of semi-feral cats. Hunting in the barns and the fields, they kill and eat rodents that would otherwise spoil large parts of the grain crop.
Thus, cats may have been an important factor in strengthening the Neolithic Revolution once it took hold, but they came too late in time to be a cause of this event which marks the beginning of human history.
Other developments that one could plausibly have expected to be triggers for the Neolithic Revolution in the absence of hard evidence, came tens and hundreds of thousands of years earlier. Human ancestors tamed fire, used stone tools, used woooden spears, lived in reasonable large groupings (two or three dozen individuals) and had a global range, more than a hundred thousand years before the Neaderthals evolved. Painting at least pre-date modern humans. There are signs that even Neanderthals had some amount of functional language, and certainly all modern humans had full fledged spoken languages.